↑

Properties

Label 1-311-311.165-r1-0-0
Degree $1$
Conductor $311$
Sign $0.0565 - 0.998i$
Analytic cond. $33.4215$
Root an. cond. $33.4215$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.688 + 0.724i)2-s + (0.688 − 0.724i)3-s + (−0.0506 + 0.998i)4-s + (0.979 − 0.201i)5-s + 6-s + (−0.954 − 0.299i)7-s + (−0.758 + 0.651i)8-s + (−0.0506 − 0.998i)9-s + (0.820 + 0.571i)10-s + (−0.918 − 0.394i)11-s + (0.688 + 0.724i)12-s + (−0.0506 − 0.998i)13-s + (−0.440 − 0.897i)14-s + (0.528 − 0.848i)15-s + (−0.994 − 0.101i)16-s + (−0.820 − 0.571i)17-s + ⋯
L(s)  = 1  + (0.688 + 0.724i)2-s + (0.688 − 0.724i)3-s + (−0.0506 + 0.998i)4-s + (0.979 − 0.201i)5-s + 6-s + (−0.954 − 0.299i)7-s + (−0.758 + 0.651i)8-s + (−0.0506 − 0.998i)9-s + (0.820 + 0.571i)10-s + (−0.918 − 0.394i)11-s + (0.688 + 0.724i)12-s + (−0.0506 − 0.998i)13-s + (−0.440 − 0.897i)14-s + (0.528 − 0.848i)15-s + (−0.994 − 0.101i)16-s + (−0.820 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0565 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0565 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.0565 - 0.998i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.0565 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.742046027 - 1.646093137i\)
\(L(\frac12)\) \(\approx\) \(1.742046027 - 1.646093137i\)
\(L(1)\) \(\approx\) \(1.618269421 - 0.1342370709i\)
\(L(1)\) \(\approx\) \(1.618269421 - 0.1342370709i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.688 + 0.724i)T \)
3 \( 1 + (0.688 - 0.724i)T \)
5 \( 1 + (0.979 - 0.201i)T \)
7 \( 1 + (-0.954 - 0.299i)T \)
11 \( 1 + (-0.918 - 0.394i)T \)
13 \( 1 + (-0.0506 - 0.998i)T \)
17 \( 1 + (-0.820 - 0.571i)T \)
19 \( 1 + (-0.979 - 0.201i)T \)
23 \( 1 + (0.758 - 0.651i)T \)
29 \( 1 + (0.440 - 0.897i)T \)
31 \( 1 + (-0.820 + 0.571i)T \)
37 \( 1 + (0.954 - 0.299i)T \)
41 \( 1 + (0.250 + 0.968i)T \)
43 \( 1 + (0.954 + 0.299i)T \)
47 \( 1 + (-0.440 + 0.897i)T \)
53 \( 1 + (-0.954 - 0.299i)T \)
59 \( 1 + (0.954 + 0.299i)T \)
61 \( 1 + (-0.979 - 0.201i)T \)
67 \( 1 + (-0.250 + 0.968i)T \)
71 \( 1 + (0.612 - 0.790i)T \)
73 \( 1 + (0.151 - 0.988i)T \)
79 \( 1 + (-0.250 - 0.968i)T \)
83 \( 1 + (0.528 + 0.848i)T \)
89 \( 1 + (-0.954 + 0.299i)T \)
97 \( 1 + (0.612 - 0.790i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.55968112700289672600981722261, −24.274687866973200706103686710359, −23.20476939990393420543692475772, −22.16099523940121868639980445709, −21.6008995430054450164756538094, −21.01543269885080404082707719866, −20.0191104622566623204496787005, −19.15830397654806744552756245977, −18.40911015372489161919837836478, −16.962221683820162329120184185204, −15.80366866575045991867156431656, −15.02704431242369889531943452723, −14.16162917308624259811986430153, −13.196961095466930293760737073281, −12.743880301710076496302070487960, −11.05541431273809981275024392017, −10.34836985893749565727045098954, −9.483607389169719379053469567157, −8.86779888742186395016469732364, −6.921994975670439118935218384333, −5.855007488480452695771818227875, −4.82334949175037968257293536861, −3.73014549080963474174333725600, −2.58981742717746583055796997469, −1.96243565810854238920017463820, 0.44843736718068453478041273626, 2.493704908930935231058952097087, 3.03562809027860185334098058077, 4.59942379861015539919293699007, 5.92389539578263749707110965303, 6.54105695050760688167022215084, 7.60923497851743271763428751368, 8.61483785769817087095180149403, 9.515464747843062295575264021768, 10.869349825421813257728536120711, 12.69627254378958700875274567069, 12.96015177499090470442834462943, 13.608411348566470317852668629519, 14.59933302736584569236632740885, 15.569761431783170687414038749742, 16.52568233638524356351969505019, 17.608696780846954659295963062279, 18.22361307298582897255762241134, 19.47699880383206155646873197887, 20.5374003592825538586282819254, 21.196212184256308521265320554683, 22.307320773987680308402264032458, 23.157081867415976125193182975785, 24.02384768029997456525858394251, 25.0065777590879546854855079667

Graph of the $Z$-function along the critical line